Mathematical discussion: Incoherent exchanges or shared understandings?

Abstract The data examined in this paper are from one group of those collected for a project addressing the question of whether pupil‐pupil discussion aids mathematical understanding. They represent episodes which were initially coded as ‘incoherent’ by the observers, but which on closer inspection appeared, for the participants, to be relevant to the mathematical discussion taking place. A variety of features, which lead to this external appearance of incoherence, were revealed through deeper analysis and the key phenomenon seemed to be that of unarticulated, but shared, meanings. One of the major problems for the listener occurs when the shared meanings are in fact in some way erroneous. This can manifest itself in misuse of mathematical language or invention of personal vocabularies, in reliance on memories of earlier, possibly misconstrued, events and in reference to shared, previously created, visual images for which they do not need explicit language. The features which this paper exposes give rise to some pertinent questions about the relationship between pupil‐pupil discussion and mathematical learning. The teacher's concern must be to extract the true factors inhibiting understanding from those appearing on the surface of the discussion.